On the Kelvin-Helmholtz instability in superfluids

The Kelvin-Helmholtz instability in superfluids is discussed on the basis of the first experimental observation of such an instability at the interface between superfluid ³He-A and superfluid ³He-B (R. Blaauwgeers, V. B. Eltsov, G. Eska et al., cond-mat/0111343). We discuss why the Kelvin-Helmholtz criterion, the Landau critical velocity for nucleation of ripplons, and the free-energy consideration all give different values for the instability threshold.     

Kelvin-Helmholtz instability of miscible ferrofluids

We study the stability of an interface between two inviscid magnetic fluids of different densities flowing parallel to each other in an oscillatory manner. The system is pervaded by a uniform oblique magnetic field distribution. The analysis allows for mass and heat transfer across the interface. A general eigenvalue relation is derived and discussed analytically. The classical stability criterion is found to be substantially modified due to the effect of the oblique magnetic field with mass and heat transfer. Some previous studies are reported for appropriate data choices. The longitudinal magnetic field has a strong stabilizing influence on all wavelengths, which can be used to suppress the destabilizing influence of the mass and heat transfer. We conclude with a discussion of the stability of unsteady shear layers on the basis of the results. The parametric excitation of the surface waves is analyzed by means of the multiple-time-scales method. The transition curves are obtained analytically.     

On stochastic mixing caused by the Rayleigh-Taylor instability

The mixing of contacting substances is considered. The evolution of the mixing layer over a long time period from multimode initial perturbations is investigated numerically in the short-scale and wide-range cases. In the case of a short-scale initiation, the flow is stochastic in the sense that the time of the considered evolution exceeds the period of correlation. The effect of the amplitude of wide-range perturbations on the dynamics of mixing is analyzed. The scale-invariant properties of the spectral and statistical parameters of turbulent mixing are investigated for the first time. The universal spectra characterizing the turbulence mixing in the entire self-similar interval on a unified basis are obtained. The simulation is based on the effective algorithms with high approximating qualities, which have been tested earlier.     

Nonlinear asymptotic calculation of the Kelvin-Helmholtz instability

The problem of periodic capillary-gravitational wave motion on the uniformly charged interface between two ideal immiscible incompressible liquids is solved in the third order of smallness. The lower liquid is assumed to be ideally conducting, while the upper one is a dielectric executing translational motion parallel to the interface with a constant velocity. A nonlinear frequency correction in the resonance form is found. It is shown that the positions of internal nonlinear resonances depend on the sum of the field and Weber parameters, the density ratio of the liquids, and the wave number. When the upper liquid is denser than the lower one, resonances are absent.     

Magnetic reconnection in the two-dimensional Kelvin-Helmholtz instability

Magnetic reconnection in the two-dimensional Kelvin-Helmholtz instability is studied. The flow is modeled by the reduced MHD equations with constant resistivity and viscosity. For super-Alfvénic flow, localized transient reconnection is observed on the Kelvin-Helmholtz time scale (this is not new). We study this transient reconnection and consider the peak reconnection rate which occurs with the initial vortex formation. Over the range of resistivities considered, it is shown that this peak reconnection rate is not a function of resistivity, and is a function of the initial flow shear. Additionally, it is demonstrated that there is a fundamental difference between the evolution of a problem at S = 200 and S = 10,000.     

Self-similar wave produced by local perturbation of the Kelvin-Helmholtz shear-layer instability

We show that the Kelvin-Helmholtz instability excited by a localized perturbation yields a self-similar wave. The instability of the mixing layer was first conceived by Helmholtz as the inevitable growth of any localized irregularity into a spiral, but the search and uncovering of the resulting self-similar evolution was hindered by the technical success of Kelvin's wavelike perturbation theory. The identification of a self-similar solution is useful since its specific structure is witness of a subtle nonlinear equilibrium among the forces involved. By simulating numerically the Navier-Stokes equations, we analyze the properties of the wave: growth rate, propagation speed and the dependency of its shape upon the density ratio of the two phases of the mixing layer.     

二维不可压缩流体Kelvin-Helmholtz不稳定性的二次谐波产生

推导了Kelvin-Helmholtz(KH)不稳定性二次谐波产生的理论公式,利用该公式得到的二次谐波增长的弱非线性结果与LARED-S程序的模拟结果符合.二维模拟定出了KH不稳定性单模非线性增长的阈值. 关键词： Kelvin-Helmholtz不稳定性 二次谐波产生 非线性阈值     

Modulational instability of dark solitons in three wave resonant interaction

We study the propagation of velocity-locked dark triplet solitons in the three wave resonant interaction model. The modulational instability of the plane wave background where the solitons sit prevents the long range propagation. However even a small second order dispersion proves to greatly reduce, or suppress, the modulational instability gain, allowing for effective stable soliton propagation.     

Integrability of double three-wave interaction

We consider the nonlinear interaction of four coherent waves with two of them participating simultaneously in two resonant triplets. Integrability of this system is revealed by the discovery of a fourth integral of the motion, to add to the three already known (the hamiltonian and the two Manley-Rowe relations).     

Subharmonic suppression of a transverse Kelvin-Helmholtz instability

A plasma density modulation applied at nearly twice the frequency of an unstable mode was observed to produce amplitude suppression exhibiting strong hysteresis.     

双界面耦合作用对KH不稳定的影响（英）

研究了含有中间层的流体系统的双界面KH不稳定性。计算发现:由于耦合作用,界面KH不稳定性呈现出不同的增长模式。主要研究了对增长率影响最大的夹层厚度对增长模式的影响,发现厚度小于半个扰动波长时,耦合作用必须考虑。另外发现在特定厚度下,界面不稳定性将出现简并的增长模式。     

双界面耦合作用对KH不稳定的影响（英）

研究了含有中间层的流体系统的双界面KH不稳定性。计算发现：由于耦合作用,界面KH不稳定性呈现出不同的增长模式。主要研究了对增长率影响最大的夹层厚度对增长模式的影响,发现厚度小于半个扰动波长时,耦合作用必须考虑。另外发现在特定厚度下,界面不稳定性将出现简并的增长模式。     

Propagative Sine-Gordon solitons in the spatially forced Kelvin-Helmholtz instability

The spatio-temporal evolution of the vortex sheet separating two finite-depth layers of immiscible fluids is examined in the vicinity of threshold when spatially periodic forcing is imposed at the horizontal boundaries. As a result of the Galilean invariance of the problem, the interface deformation is shown to satisfy a coupled system of evolution equations involving not only the usual “short-wave” at the critical wavenumber but also a shallow-water “long-wave” associated with the mean elevation of the interface. The weakly nonlinear model is further studied in the Boussinesq approximation where it reduces to a forced Klein-Gordon equation. Thus, the secondary Benjamin-Feir instability of nonlinear Stokes wavetrains is analysed in the absence of forcing. When spatial forcing is reintroduced, the competition between the imposed external length scale and the natural length scale of the interface is shown analytically to give rise to one-dimensional propagating Sine-Gordon phase solitons. Numerical simulations of the Klein-Gordon evolution model fully confirm this prediction and also lead to the determination of the range of stability of phase solitons.     

二维不可压流体Kelvin-Helmholtz不稳定性的弱非线性研究

通过将扰动速度势展至三阶,提出了Kelvin-Helmholtz（KH）不稳定性的弱非线性理论.在模耦合过程中观察到一个重要的共振现象,共振使得模耦合过程变得相当复杂,单模扰动很快进入非线性区,产生大量高次谐波,共振加强了非线性作用.分析了单模扰动中二次和三次谐波产生效应,以及对基模指数增长的非线性校正.模拟结果支持了解析理论.利用该理论,分析了KH不稳定的非线性阈值问题. 关键词： Kelvin-Helmholtz不稳定性 弱非线性理论 非线性阈值     

On the resonant interaction of spherical ion-acoustic solitons

The resonant interaction of two spherical ion-acoustic solitons is observed. A new soliton whose amplitude is approximately three times the colliding soliton's amplitude is created. The spatial growth and decay of the enhancement is noted.     

Parametric buildup of the instability of a charged flat liquid surface imposed on Kelvin-Helmholtz instability

The instability of capillary gravitational waves that is developed at the charged flat interface between media is studied for the case when the upper medium moves parallel to the interface with a velocity that has constant and time-dependent components. It is shown that the Mathieu-Hill equation, describing the temporal evolution of the capillary wave amplitudes in such a system, has unstable solutions at those values of physical parameters (electric field strength and wind velocity) meeting the conditions for Saint Elmo fire initiation in the atmosphere.